Let a polygon or polyhedron Ω, a function v in the Sobolev space H1(Ω), and a simplicial mesh of Ω be given. We prove the equivalence of two piecewise polynomial best approximations of v: 1) globally on the whole computational domain Ω, with the (trace) continuity requirement; 2) locally on each mesh element, without any interelement continuity requirement. The former (global-best continuous piecewise polynomial approximation) arises in numerical methods for partial differential equations related to the H1(Ω) space, whereas the latter (local-best discontinuous piecewise polynomial approximation) is a key quantity in approximation theory. Crucially, we establish p-robustness in that the equivalence constant only depends on the mesh shape regularity and the spatial dimension. This improves the recent results of [Found. Comput. Math. 16 (2016), 723-750] and [Numer. Math. 135 (2017), 1073-1119], where the equivalence constant was possibly dependent (algebraically or logarithmically) on the underlying polynomial degree. Consequently, we obtain fully h- and p- (mesh-size and polynomial-degree-) optimal approximation estimates under the minimal Sobolev regularity only requested separately on each mesh element, where we also cover locally variable polynomial degrees. These two results immediately follow by our construction of an operator from the infinite-dimensional Sobolev space H1(Ω) to its finite-dimensional piecewise polynomial subspace that has the following properties: 1) it is defined over the entire H1(Ω) and preserves boundary conditions imposed on a part of the boundary of Ω; 2) it is defined locally in a neighborhood of each mesh element; 3) it is based on elementwise H1-orthogonal polynomial projections; 4) it is a projector, i.e., it leaves intact objects that are already continuous piecewise polynomials; 5) it is locally and p-robustly stable in the H1(Ω)-seminorm; 6) its approximation property is locally and p-robustly equivalent to that of the local discontinuous (elementwise H1-orthogonal) projection.